1. Field of the Invention
The present invention relates to an equalizer used in a communication network, and more specifically, to a method and apparatus for simplifying the metric calculation of Maximum Likelihood Sequence Estimation (MLSE).
2. Description of the Related Art
The modulation algorithm employed in both Global System for Mobile (GSM) and General Packet Radio System (GPRS) communication networks is Gaussian Minimum Shift Keying (GMSK), which induces inter-symbol interference (ISI) in the received signal sampled at a specific rate. ISI is the residual effect of other neighboring symbols when decoding a certain symbol and this residual effect is due to the occurrence of pulses before and after the sampling instance. The unavoidable presence of ISI in the system, however, introduces errors in the decision device at the receiver output. Therefore, in the filter design of the GSM/GPRS receiver and transmitter, the object is to diminish the effects of ISI and thereby deliver the digital data to the destination with the smallest error rate possible. An equalizer is a widely used approach for compensating or reducing the ISI effect.
In general, there are three types of equalizers: maximum likelihood (ML) equalizer, liner equalizer, and decision feedback equalizer. The ML equalizer is based on the maximum likelihood sequence detection criterion, whereas the linear equalizer is based on the use of a linear filter with adjustable coefficients. The decision feedback equalizer is based on the use of previous detected symbols to suppress the ISI in the present symbol being detected. The performance and complexity of these equalizers are listed in the table shown in FIG. 1. The ML equalizer has optimum performance among the three equalizers. The complexity of the ML equalizer, however, is the highest. In FIG. 1, L indicates the length of ISI.
An exemplary transmission model of a wireless communication system is shown in FIG. 2. A signal 21 from the transmitter 22 is first filtered by a Low Pass (LP) filter 221, converted to radio frequency (RF) by multiplying a carrier in a multiplier 222, and finally passed to a processing unit 223 before transmission to the channel 24. The processing unit 223 extracts the real part of the signal. The characteristic of the channel 24 is modeled by a channel response block 241 with channel noise 23. The channel noise 23 is generally assumed as Additive White Gaussian Noise (AWGN). The multiplier 261 in the receiver 26 converts back the frequency of the signal received by the receiver by multiplying the same frequency as the carrier frequency. The LP filter 262 receives the down converted signal, and outputs a received signal 25. The equalizer 263 compensates the ISI of the received signal 25, and outputs an estimated signal 27. The received signal r(t) 25 is expressed by Equation [1].
                              r          ⁡                      (            t            )                          =                                            ∑              n                        ⁢                                                  ⁢                                          a                n                            ⁢                              h                ⁡                                  (                                      t                    -                    nT                                    )                                                              +                      n            ⁡                          (              t              )                                                          Equation        ⁢                                  [        1        ]            
h(t) denotes the overall channel response of the system, and it can be derived by performing convolution to the transmitter's impulse response ft(f), receiver's impulse response fr(t), and channel model response g(t), h(t)=ft(t)*g(t)*fr(t).
The channel noise n(t) 23 herein is assumed to be stationary Gaussian noise with zero mean and variance No. Let {an} (which is also the signal 21 in FIG. 2) be a hypothetical sequence of pulse amplitudes transmitted during a time period I. The equalizer 263 is assumed to be a ML equalizer, which determines the best estimation of {an} as the estimated sequence {ân} (signal 27 of FIG. 2). The estimated sequence {αn}={ân} is derived by maximizing the likelihood function as shown in Equation [2.p[r(t),t ε I |{αn}]  Equation [2]
The probability of 0s and 1s in the transmitted sequences are assumed to be equal, therefore, Equation [2] can be rewritten as:
                              p          ⁡                      [                                          {                                  α                  n                                }                            ❘                              r                ⁡                                  (                  t                  )                                                      ]                          =                                            p              ⁡                              (                                                      r                    ⁡                                          (                      t                      )                                                        ❘                                      {                                          α                      n                                        }                                                  )                                      ⁢                          p              ⁡                              (                                  {                                      α                    n                                    }                                )                                                          p            ⁡                          (                              r                ⁡                                  (                  t                  )                                            )                                                          Equation        ⁢                                  [        3        ]            
p[{αn}|r(t)] is also called the posteriori probability. The probability of the estimated sequence p[{αn}] and the received signal r(t) are both assumed to be constant. Since the objective of the ML equalizer is to maximize the likelihood function shown in Equation [2], the posteriori probability must also be maximized. If the sequence {αn} was the actual sequence of the pulse amplitude transmitted during time period I, the power density function of the noise signal n(t) can be expressed as shown in Equation [4].
                                          n            ⁡                          (                              t                ❘                                  {                                      α                    n                                    }                                            )                                =                                    r              ⁡                              (                t                )                                      -                                          ∑                                  nT                  ∈                  I                                            ⁢                                                          ⁢                                                α                  n                                ⁢                                  h                  ⁡                                      (                                          t                      -                      nT                                        )                                                                                      ,                  t          ∈          I                                    Equation        ⁢                                  [        4        ]            
The ML function of Equation [2] thus becomes:
                                                                        p                ⁡                                  (                                                            r                      ⁡                                              (                        t                        )                                                              ❘                                          {                                              α                        n                                            }                                                        )                                            =                              p                ⁡                                  [                                                            n                      ⁡                                              (                        t                        )                                                              ❘                                          {                                              α                        n                                            }                                                        ]                                                                                                        =                                                                    (                                          1                                              2                        ⁢                        π                        ⁢                                                                                                  ⁢                                                  N                          o                                                                                      )                                    N                                ⁢                                  exp                  ⁡                                      (                                                                                                                                                                                      -                                                                  1                                                                      2                                    ⁢                                                                          N                                      o                                                                                                                                                                  ⁢                                                              ∑                                                                  k                                  =                                  1                                                                N                                                                                      ⁢                                                                                                                                                                                                                                                                                                                                                                                                r                                  k                                                                -                                                                                                      ∑                                    n                                                                    ⁢                                                                                                                                          ⁢                                                                                                            α                                      n                                                                        ⁢                                                                          h                                      kn                                                                                                                                                                                                                          2                                                                                                                )                                                                                                          Equation        ⁢                                  [        5        ]            
The probability of the ML function p[r(t)|{αn}] is proportional to the logarithm. p[r(t)|{αn}]:
                              -                                    ∫                              -                ∞                            ∞                        ⁢                                                                                                                        r                      ⁡                                              (                        t                        )                                                              -                                                                  ∑                        n                                            ⁢                                                                                          ⁢                                                                        α                          n                                                ⁢                                                  h                          ⁡                                                      (                                                          t                              -                              nT                                                        )                                                                                                                                                                2                            ⁢                                                          ⁢                              ⅆ                t                                                    =                              -                                          ∫                                  -                  ∞                                ∞                            ⁢                                                                                                              r                      ⁡                                              (                        t                        )                                                                                                  2                                ⁢                                  ⅆ                  t                                                              +                      2            ⁢            Re            ⁢                                          ∑                n                            ⁢                                                          ⁢                              [                                                      α                    n                    *                                    ⁢                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                  r                        ⁡                                                  (                          t                          )                                                                    ⁢                                                                        h                          *                                                ⁡                                                  (                                                      t                            -                            nT                                                    )                                                                    ⁢                                              ⅆ                        t                                                                                            ]                                              -                                    ∑              n                        ⁢                                                  ⁢                                          ∑                m                            ⁢                                                          ⁢                                                α                  n                  *                                ⁢                                  α                  m                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                                                    h                        *                                            ⁡                                              (                                                  t                          -                          nT                                                )                                                              ⁢                                          h                      ⁡                                              (                                                  t                          -                          mT                                                )                                                              ⁢                                                                                  ⁢                                          ⅆ                      t                                                                                                                              Equation        ⁢                                  [        6        ]            
The first term of Equation [6] is a constant, thus it can be discarded when calculating the metric. A correlation metric (MC) can be derived from the previous steps as shown in Equation [7].
                                          CM            ⁡                          (                              {                                  α                  I                                }                            )                                =                                    2              ⁢              Re              ⁢                                                ∑                                      nT                    ∈                    I                                                  ⁢                                                                  ⁢                                  (                                                            α                      n                      *                                        ⁢                                          Z                      n                                                        )                                                      -                                          ∑                                  iT                  ∈                  I                                            ⁢                                                          ⁢                                                ∑                                      kT                    ∈                    I                                                  ⁢                                                                  ⁢                                                      α                    n                    *                                    ⁢                                      α                    k                                    ⁢                                      s                                          i                      -                      k                                                                                                          ⁢                                  ⁢        where        ⁢                                  ⁢                              z            n                    =                                                                                          g                    MF                                    ⁡                                      (                    t                    )                                                  *                                  r                  ⁡                                      (                    t                    )                                                              ⁢                              ❘                                  t                  =                  nT                                                      =                                                            ∑                  l                                ⁢                                                                  ⁢                                                      a                                          n                      -                      l                                                        ⁢                                      s                    l                                                              +                              w                n                                                    ⁢                                  ⁢                              s            l                    =                                                                                          g                    MF                                    ⁡                                      (                    t                    )                                                  *                                  h                  ⁡                                      (                    t                    )                                                              ⁢                              ❘                                  t                  =                  lT                                                      =                                          s                                  -                  l                                *                            ⁢                                                          ⁢              and                                      ⁢                                  ⁢                                            g              MF                        ⁡                          (              t              )                                =                                    h              *                        ⁡                          (                              -                t                            )                                                          Equation        ⁢                                  [        7        ]            
s1 herein is the channel response autocorrelation.
Maximum Likelihood Sequence Estimation (MLSE) determines the most likely sequence originally transmitted by the sequence {αn} by maximizing the likelihood function shown in Equation [5], or equivalently, maximizing the metric shown in Equation [8].
                                          J            n                    ⁡                      (                          {                              α                I                            }                        )                          =                              2            ⁢            Re            ⁢                                          ∑                                  nT                  ∈                  I                                            ⁢                                                          ⁢                              (                                                      α                    n                    *                                    ⁢                                      Z                    n                                                  )                                              -                                    ∑                              iT                ∈                I                                      ⁢                                                  ⁢                                          ∑                                  kT                  ∈                  I                                            ⁢                                                          ⁢                                                α                  i                  *                                ⁢                                  s                                      i                    -                    k                                                  ⁢                                  α                  k                                                                                        Equation        ⁢                                  [        8        ]            
The MLSE algorithm obtained represents a modified version of the well-known Viterbi algorithm. The Viterbi algorithm is obtained by computing the recursive relation iteratively.
                                          J            n                    (                                          ⁢                      …            ⁢                                                  ,                          α                              n                =                1                                      ,                          α              n                                )                =                                            J                              n                -                1                                      (                                                  ⁢                          …              ⁢                                                          ,                              α                                  n                  -                  1                                                      )                    +                                    Re              [                                                α                  n                  *                                ⁡                                  (                                                            2                      ⁢                                              Z                        n                                                              -                                                                  s                        0                                            ⁢                                              α                        n                                                              -                                          2                      ⁢                                                                        ∑                                                      k                            ≤                                                          n                              -                              1                                                                                                      ⁢                                                                                                  ⁢                                                                              s                                                          n                              -                              k                                                                                ⁢                                                      α                            k                                                                                                                                )                                            ]                        .                                              Equation        ⁢                                  [        9        ]            
FIG. 3 shows the architecture of the Viterbi Equalizer, wherein the received signal r(t) is estimated according to Equation [1].